Mathematical Modeling

• Autore: Kapur J. N
• Pubblicazione: Bloomfield : , : Mercury Learning & Information, , 2023; ©2023
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (287 pages)
• Disciplina: 511.8
• Soggetto non controllato: Algebras, Linear; Mathematical Optimization; Difference Equations; Mathematics
• Nota di contenuto: Cover -- Title Page -- Copyright -- Contents -- Preface -- Chapter 1: Mathematical Modeling: Need, Techniques, Classifications, and Simple Illustrations -- 1.1 Simple Situations Requiring Mathematical Modeling -- 1.2 The Technique of Mathematical Modeling -- 1.3 Classification of Mathematical Models -- 1.4 Some Characteristics of Mathematical Models -- 1.5 Mathematical Modeling through Geometry -- 1.6 Mathematical Modeling through Algebra -- 1.7 Mathematical Modeling through Trigonometry -- 1.7.1 Finding the Distance of the Moon -- 1.7.2 Finding the Distance of a Star -- 1.7.3 Finding Length of the Day -- 1.8 Mathematical Modeling through Calculus -- 1.8.1 Law of Reflection -- 1.8.2 Law of Refraction of Light -- 1.8.3 EOQ Model for Inventory Control -- 1.8.4 Triangle of Given Perimeter with Maximum Area -- 1.8.5 Parallelopiped with Given Perimeter and Maximum Volume -- 1.8.6 Mathematics of Business -- 1.9 Limitations of Mathematical Modeling -- Bibliography -- Chapter 2: Mathematical Modeling through Ordinary Differential Equations of the First Order -- 2.1 Mathematical Modeling through Differential Equations -- 2.2 Linear Growth and Decay Models -- 2.2.1 Populational Growth Models -- 2.2.2 Growth of Science and Scientists -- 2.2.3 Effects of Immigration and Emigration on Population Size -- 2.2.4 Interest Compounded Continuously -- 2.2.5 Radioactive Decay -- 2.2.6 Decrease of Temperature -- 2.2.7 Diffusion -- 2.2.8 Change of Price of a Commodity -- 2.3 Nonlinear Growth and Decay Models -- 2.3.1 Logistic Law of Population Growth -- 2.3.2 Spread of Technological Innovations and Infectious Diseases -- 2.3.3 Rate of Dissolution -- 2.3.4 Law of Mass Action: Chemical Reactions -- 2.4 Compartment Models -- 2.4.1 A Simple Compartment Model -- 2.4.2 Diffusion of Glucose or a Medicine in the Bloodstream -- 2.4.3 The Case of a Succession of Compartments. ; 2.5 Mathematical Modeling In Dynamics through Ordinary Differential Equations of the First Order -- 2.5.1 Simple Harmonic Motion -- 2.5.2 Motion Under Gravity in a Resisting Medium -- 2.5.3 Motion of a Rocket -- 2.6 Mathematical Modeling of Geometrical Problems through Ordinary Differential Equations of the First Order -- 2.6.1 Simple Geometrical Problems -- 2.6.2 Orthogonal Trajectories -- Chapter 3: Mathematical Modeling through Systems of Ordinary Differential Equations of the First Order -- 3.1 Mathematical Modeling In Population Dynamics -- 3.1.1 Prey-Predator Models -- 3.1.2 Competition Models -- 3.1.3 Multispecies Models -- 3.1.4 Age-Structured Population Models -- 3.2 Mathematical Modeling of Epidemics through Systems of Ordinary Differential Equations of the First Order -- 3.2.1 A Simple Epidemic Model -- 3.2.2 A Susceptible-Infected-Susceptible (SIS) Model -- 3.2.3 SIS Model with Constant Number of Carriers -- 3.2.4 Simple Epidemic Model with Carriers -- 3.2.5 Model with Removal -- 3.2.6 Model with Removal and Immigration -- 3.3 Compartment Models through Systems of Ordinary Differential Equations -- 3.4 Mathematical Modeling In Economics Based On Systems of Ordinary Differential Equations of the First Order -- 3.4.1 Domar Macro Model -- 3.4.2 Domar First Debt Model -- 3.4.3 Domar's Second Debt Model -- 3.4.4 Allen's Speculative Model -- 3.4.5 Samuelson's Investment Model -- 3.4.6 Samuelson's Modified Investment Model -- 3.4.7 Stability of Market Equilibrium -- 3.4.8 Leontief's Open and Closed Dynamical Systems for Inter-Industry Relations -- 3.5 Mathematical Models in Medicine, Arms Race Battles, and International Trade In Terms of Systems of Ordinary Differential Equations -- 3.5.1 A Model for Diabetes Mellitus -- 3.5.2 Richardson's Model for the Arms Race -- 3.5.3 Lanchester's Combat Model -- 3.5.4 International Trade Model. ; 3.6 Mathematical Modeling In Dynamics through Systems of Ordinary Differential Equations of the First Order -- 3.6.1 Modeling in Dynamics -- 3.6.2 Motion of a Projectile -- 3.6.3 External Ballistics of Gun Shells -- Chapter 4: Mathematical Modeling through Ordinary Differential Equations of the Second Order -- 4.1 Mathematical Modeling of Planetary Motions -- 4.1.1 Need for the Study of Motion Under Central Forces -- 4.1.2 Components of Velocity and Acceleration Vectors along Radial and Transverse Directions -- 4.1.3 Motion Under a Central Force -- 4.1.4 Motion Under the Inverse Square Law -- 4.1.5 Kepler's Laws of Planetary Motions -- 4.2 Mathematical Modeling of Circular Motion and Motion of Satellites -- 4.2.1 Circular Motion -- 4.2.2 Motion of a Particle on a Smooth or Rough Vertical Wire -- 4.2.3 Circular Motion of Satellites -- 4.2.4 Elliptic Motion of Satellites -- 4.3 Mathematical Modeling through Linear Differential Equations of the Second Order -- 4.3.1 Rectilinear Motion -- 4.3.2 Electrical Circuits -- 4.3.3 Phillip's Stabilization Model for a Closed Economy -- 4.4 Miscellaneous Mathematical Models through Ordinary Differential Equations of the Second Order -- 4.4.1 The Catenary -- 4.4.2 A Curve of Pursuit -- Chapter 5: Mathematical Modeling through Difference Equations -- 5.1 The Need For Mathematical Modeling through Difference Equations: Some Simple Models -- 5.2 Basic Theory of Linear Difference Equations With Constant Coefficients -- 5.2.1 The Linear Difference Equation -- 5.2.2 The Complementary Function -- 5.2.3 The Particular Solution -- 5.2.4 Obtaining the Complementary Function by Use of Matrices -- 5.2.5 Solution of a System of Linear Homogeneous Difference Equations with Constant Coefficients -- 5.2.6 Solution of Linear Difference Equations by Using the Laplace Transform. ; 5.2.7 Solution of Linear Difference Equations by Using the z-Transform -- 5.2.8 Solution of Nonlinear Difference Equations Reducible to Linear Equations -- 5.2.9 Stability Theory for Difference Equations -- 5.3 Mathematical Modeling through Difference Equations In Economics And Finance -- 5.3.1 The Harrod Model -- 5.3.2 The Cobweb Model -- 5.3.3 Samuelson's Interaction Models -- 5.3.4 Application to Actuarial Science -- 5.4 Mathematical Modeling through Difference Equations In Population Dynamics and Genetics -- 5.4.1 Nonlinear Difference Equations Model for Population Growth: Nonlinear Difference Equations -- 5.4.2 Age-Structured Population Models -- 5.4.3 Mathematical Modeling through Difference Equations in Genetics -- 5.5 Mathematical Modeling through Difference Equations In Probability Theory -- 5.5.1 Markov Chains -- 5.5.2 Gambler's Ruin Problems -- 5.6 Miscellaneous Examples of Mathematical Modeling through Difference Equations -- Chapter 6: Mathematical Modeling through Partial Differential Equations -- 6.1 Situations Giving Rise To Partial Differential Equation Models -- 6.2 Mass Balance Equations: First Method of Getting PDE Models -- 6.2.1 Equation of Continuity in Fluid Dynamics -- 6.2.2 Equation of Continuity for Heat Flow -- 6.2.3 Equation of Continuity for Traffic Flow on a Highway -- 6.2.4 Gauss Divergence Theorem in Electrostatics -- 6.2.5 Mathematical Modeling in Terms of Laplace's Equation -- 6.2.6 Mathematical Modeling in Terms of the Diffusion Equation -- 6.3 Momentum Balance Equations: the Second Method of Obtaining Partial Differential Equation Models -- 6.3.1 Euler's Equations of Motion for Inviscid Fluid Flow -- 6.3.2 Partial Differential Equation Model for a Vibrating String -- 6.3.3 Partial Differential Equation Model for a Vibrating Membrane -- 6.3.4 Mathematical Modeling in Terms of the Wave Equation. ; 6.4 Variational Principles: Third Method of Obtaining Partial Differential Equation Models -- 6.4.1 Euler-Lagrange Equation -- 6.4.2 Minimal Surfaces -- 6.4.3 Vibrating String -- 6.4.4 Vibrating Membrane -- 6.4.5 Gas Filled Cylinder -- 6.5 Probability Generating Function, Fourth Method of Obtaining Partial Differential Equation Models -- 6.5.1 PDE Model for Birth-Death-Immigration-Emigration Process -- 6.5.2 PDE Model for a Stochastic Epidemic Process with No Removal -- 6.5.3 Stochastic Epidemic Model with No Removal -- 6.6 Model For Traffic On A Highway -- 6.6.1 Relation Between Car Velocity U and Traffic Density ρ -- 6.6.2 An Alternative Relation Between U and ρ -- 6.6.3 Traffic Wave Propagation Along a Highway -- 6.7 Nature of Partial Differential Equations -- 6.7.1 Elliptic, Parabolic, and Hyperbolic Equations -- 6.7.2 Nature of Three Basic Linear Partial Differential Equations -- 6.7.3 The Nature of the Partial Differential Equation for the Potential of the Steady Two-Dimensional Flow of the Inviscid Flow of an Ideal Gas -- 6.8 Initial and Boundary Conditions -- Chapter 7: Mathematical Modeling through Graphs -- 7.1 Situations That Can Be Modeled through Graphs -- 7.1.1 Qualitative Relations in Applied Mathematics -- 7.1.2 The Seven Bridges Problem -- 7.1.3 Some Types of Graphs -- 7.1.4 Nature of Models in Terms of Graphs -- 7.2 Mathematical Models In Terms of Directed Graphs -- 7.2.1 Representing Results of Tournaments -- 7.2.2 One-Way Traffic Problems -- 7.2.3 Genetic Graphs -- 7.2.4 Senior-Subordinate Relationship -- 7.2.5 Food Webs -- 7.2.6 Communication Networks -- 7.2.7 Matrices Associated with a Directed Graph -- 7.2.8 Application of Directed Graphs to Detection of Cliques -- 7.3 Mathematical Models In Terms of Signed Graphs -- 7.3.1 Balance of Signed Graphs -- 7.3.2 Structure Theorem and its Implications. ; 7.3.3 Antibalance and Duobalance of a Graph.
• Sommario/riassunto: This book can be used in courses on mathematical modeling at the senior undergraduate or graduate level, or used as a reference for in-service scientists and engineers. The book provides an overview of mathematical modeling through a panoramic view of applications of mathematics in science and technology. In each chapter, mathematical models are chosen from the physical, biological, social, economic, management, and engineering sciences. The models deal with differentconcepts, but have a common mathematical structure and bring out the unifying influence of mathematical modeling in different disciplines.FEATURES:Provides a balance between theory and applicationsFeatures models from the physical, biological, social, economic, management, and engineering sciences
• Titolo autorizzato: Mathematical Modeling
• ISBN: 9781683928737; 9781683928744
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910838333803321
• Lo trovi qui: Univ. Federico II